Why it matters
Is this number prime? Will it divide exactly? What is left over? Students who know the rules answer such questions in 15–20 seconds; others do long division and lose a minute.
Plain definition: The number system groups numbers into families (natural, whole, integer, rational, irrational, real) and gives the rules for how they behave. Divisibility asks, "Does one number divide another exactly, with nothing left over?" The remainder is what is left over when the division is not exact.
Where it appears in the Constable paper: Arithmetic asks direct questions like "Which is a prime number?", "Find the missing digit for divisibility by 9" and "Find the remainder when … is divided by …". These skills also power HCF–LCM, simplification and fractions.
Core concept
Level 1 — Beginner: the families of numbers
| Family | Symbol | What it contains | Examples |
|---|---|---|---|
| Natural numbers | N | Counting numbers, starting from 1 | 1, 2, 3, 4, … |
| Whole numbers | W | Natural numbers plus 0 | 0, 1, 2, 3, … |
| Integers | Z | Whole numbers plus negatives | …, −2, −1, 0, 1, 2, … |
| Rational numbers | Q | Can be written as p/q (p, q integers, q ≠ 0) | 3 (= 3/1), −5/7, 0.25, 0.333… |
| Irrational numbers | — | Numbers that cannot be written as p/q | √2, √3, √5, π |
| Real numbers | R | All rational and irrational numbers | every point on the number line |
Each family sits inside the next: N ⊂ W ⊂ Z ⊂ Q ⊂ R. So every natural number is also whole, an integer, rational and real.
Recognise them by their decimals:
- Rational: the decimal stops (3/8 = 0.375) or repeats (2/3 = 0.666…).
- Irrational: the decimal never stops and never repeats (√2 = 1.41421356…).
- A fraction in lowest terms terminates only if its denominator has no prime factors other than 2 and 5: 7/40 terminates (40 = 2³ × 5); 5/12 repeats (12 has the factor 3).
Other important groups:
- Even numbers are divisible by 2: 0, 2, 4, … (0 is even, because 0 = 2 × 0). Odd numbers leave remainder 1 when divided by 2.
- A prime has exactly two factors, 1 and itself (2, 3, 5, 7, 11, …). A composite has more than two (4, 6, 8, 9, …).
- 1 is neither prime nor composite (it has only one factor); 0 is also neither.
- 2 is the only even prime — every other even number also has 2 as a factor.
- Co-prime numbers have no common factor except 1, like 8 and 15. They need not be prime themselves.
- Twin primes are primes that differ by 2: (3, 5), (11, 13), (17, 19), (29, 31).
- There are 15 primes up to 50 and 25 primes up to 100.
How to test if a number is prime: check divisibility only by primes up to its square root. Is 173 prime? 14² = 196 > 173, so check 2, 3, 5, 7, 11, 13. None divides 173, so 173 is prime. Why? If 173 had a factor above 14, its partner factor would be below 14, and those are already checked.
Level 2 — Intermediate: divisibility rules
Why does the digit-sum rule work? 10, 100, 1000 … all leave remainder 1 when divided by 9, so a number and its digit sum leave the same remainder on division by 9 (and by 3).
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6 or 8 | 5,738 ✔ |
| 3 | Sum of digits divisible by 3 | 4,812 → 15 ✔ |
| 4 | Last two digits divisible by 4 | 7,316 → 16 ✔ |
| 5 | Last digit is 0 or 5 | 9,045 ✔ |
| 6 | Divisible by both 2 and 3 | 1,452 ✔ |
| 7 | Double the last digit, subtract from the rest; repeat until 0 or a multiple of 7 | 2,401 → 240 − 2 = 238 → 23 − 16 = 7 ✔ |
| 8 | Last three digits divisible by 8 | 41,264 → 264 = 8 × 33 ✔ |
| 9 | Sum of digits divisible by 9 | 7,281 → 18 ✔ |
| 10 | Last digit is 0 | 3,670 ✔ |
| 11 | Odd-place sum − even-place sum = 0 or a multiple of 11 | 918,082 → see below |
| 25 | Last two digits are 00, 25, 50 or 75 | 6,175 ✔ |
Rule for 11 (918,082): count places from the right. Odd places: 2 + 0 + 1 = 3. Even places: 8 + 8 + 9 = 25. Difference 22 is a multiple of 11, so 918,082 is divisible by 11.
Composite divisors — the co-prime rule: break the divisor into co-prime factors and test each: 12 = 3 × 4, 24 = 3 × 8, 36 = 4 × 9, 72 = 8 × 9.
Why co-prime? 24 is not 4 × 6 for this purpose: 12 is divisible by both 4 and 6, yet not by 24, because 4 and 6 share the factor 2. Co-prime factors never overlap, so passing both tests guarantees divisibility by the product.
Level 3 — Advanced: remainders
The division rule (division algorithm):
Dividend = Divisor × Quotient + Remainder, where 0 ≤ Remainder < Divisor.
If your "remainder" is equal to or bigger than the divisor, you have not finished dividing.
Remainder shortcuts: by 2, 5 or 10, use the last digit; by 4 or 25, the last two digits; by 8, the last three digits; by 3 or 9, the digit sum.
Remainders of sums and products: replace each number by its remainder, do the operation, then take the remainder again.
- (17 × 23 × 29) ÷ 5 → remainders 2, 3, 4 → 2 × 3 × 4 = 24 → 24 ÷ 5 leaves 4.
Remainder by a factor of the divisor: if N ÷ D leaves remainder R, and d is a factor of D, then N ÷ d leaves the same remainder as R ÷ d.
Remainders of powers:
- (x + 1)ⁿ ÷ x always leaves remainder 1.
- (x − 1)ⁿ ÷ x leaves remainder 1 if n is even, and x − 1 if n is odd.
- Otherwise, find a small power that leaves remainder 1 and use it to cut the big power down (Example 6).
- ✓- N ⊂ W ⊂ Z ⊂ Q ⊂ R. 0 is a whole number but not a natural number.
- ✓- Rational = p/q (q ≠ 0), decimal terminates or repeats. Irrational = never ends, never repeats (√2, π). 22/7 is rational — only an approximation of π.
- ✓- 1 is neither prime nor composite; 2 is the only even prime; 0 is even.
- ✓- 15 primes up to 50; 25 primes up to 100.
- ✓- Prime test: divide only by primes up to the square root.
- ✓- 2, 4, 8 → last 1, 2, 3 digits; 3 and 9 → digit sum; 11 → odd-place sum − even-place sum = 0 or multiple of 11.
- ✓- Composite divisor → split into co-prime factors (24 = 3 × 8, 72 = 8 × 9).
- ✓- Dividend = Divisor × Quotient + Remainder, remainder < divisor.
- ✓- (x + 1)ⁿ ÷ x → remainder 1; (x − 1)ⁿ ÷ x → 1 (n even) or x − 1 (n odd).
Worked examples
Example 1 (Easy) — Classification
Question: Which of the following is an irrational number? (a) √49 (b) 0.333… (c) √12 (d) 22/7
Working:
- √49 = 7, a whole number → rational.
- 0.333… = 1/3 → rational (it repeats).
- 22/7 is in p/q form → rational.
- √12 = √(4 × 3) = 2√3. Since √3 is irrational, 2√3 is irrational.
Answer: (c) √12
Example 2 (Easy) — Missing digit for 9
Question: The number 52★346 is divisible by 9. Find the digit ★.
Working:
- Sum of known digits = 5 + 2 + 3 + 4 + 6 = 20.
- The next multiple of 9 after 20 is 27, so ★ = 27 − 20 = 7.
- 36 would need ★ = 16, not a digit.
Answer: ★ = 7 (check: 527,346 → digit sum 27 ✔)
Example 3 (Medium) — Composite divisor
Question: Which number is divisible by 24? (a) 35,816 (b) 47,532 (c) 63,672 (d) 41,814
Working: 24 = 3 × 8, so test 3 (digit sum) and 8 (last three digits).
- (a) Digit sum 23 → not divisible by 3. ✘
- (b) Digit sum 21 ✔, but 532 ÷ 8 = 66.5 ✘. (It is divisible by 4 and 6 — the 4 × 6 trap.)
- (c) Digit sum 24 ✔, and 672 = 8 × 84 ✔.
- (d) 14 is not divisible by 4, so the number is not divisible by 8. ✘
Answer: (c) 63,672
Example 4 (Medium) — Missing digit for 11
Question: Find the digit x if 8x9472 is divisible by 11.
Working: count places from the right.
- Odd places (1st, 3rd, 5th): 2, 4, x → sum = 6 + x.
- Even places (2nd, 4th, 6th): 7, 9, 8 → sum = 24.
- Difference = 24 − (6 + x) = 18 − x. For a digit x, this lies between 9 and 18, so the only multiple of 11 it can be is 11: 18 − x = 11 gives x = 7.
Answer: x = 7 (check: 879,472 → difference 11 ✔)
Example 5 (Medium-hard) — Remainder by a factor
Question: A number leaves remainder 39 when divided by 357. What is the remainder when the same number is divided by 17?
Working:
- Write the number as N = 357k + 39.
- 357 = 17 × 21, so 357k is a multiple of 17 and leaves no remainder.
- Only 39 matters: 39 = 17 × 2 + 5.
Answer: 5
This works only because 17 is a factor of 357; for a non-factor divisor the answer is not fixed.
Example 6 (Exam-hard) — Remainder of a power
Question: Find the remainder when 2¹⁰⁰ is divided by 7.
Working:
- Find a small power of 2 that leaves remainder 1: 2³ = 8 = 7 + 1.
- 100 = 3 × 33 + 1, so 2¹⁰⁰ = (2³)³³ × 2.
- (2³)³³ leaves remainder 1³³ = 1; then 1 × 2 = 2.
Answer: 2
Bonus: 5²³ ÷ 6 → 5 = 6 − 1 and 23 is odd, so remainder = 5.
Real-world connection
Split 250 trainees into squads of 8: since 250 = 8 × 31 + 2, there are 31 full squads and 2 trainees left over.
Weekdays repeat every 7 days. If today is Monday, which day is 100 days later? 100 ÷ 7 leaves 2, so count 2 days forward: Wednesday. Full weeks never change the day; only the remainder does.
- Calling 1 a prime. A prime needs exactly two factors; 1 has only one. And 2 is prime; 0 is even.
- Using factors that are not co-prime. Testing 24 with 4 and 6 fails: 12 passes both but is not divisible by 24. Use 3 × 8.
- Leaving a remainder bigger than the divisor. If remainders 4 and 3 (divisor 5) are added, the sum 7 is not the answer; 7 ÷ 5 leaves 2.
- Thinking every non-terminating decimal is irrational. 0.666… repeats, so it is rational (= 2/3).
- Ending with a negative remainder. −1 for divisor 6 means the true remainder is 6 − 1 = 5.
- Cheapest rule first: last digit (2, 5, 10), then digit sum (3, 9), then last two or three digits (4, 8).
- Eliminate fast: for "divisible by 72", first cross out options whose digit sum is not a multiple of 9.
- Missing digit for 9: add the known digits and jump to the next multiple of 9. If the known sum is already a multiple of 9, both 0 and 9 work — check whether the question wants the smallest or largest.
- Powers: base one more than the divisor → remainder 1; one less → check if the power is even or odd.
- Mnemonic — "2-4-8 needs 1-2-3": for 2 check the last 1 digit, for 4 the last 2 digits, for 8 the last 3 digits. The same pattern works for 5, 25, 125.
How it is asked in the exam
You get about 54 seconds per question; finish easy number-system questions in about 20 seconds to bank time.
Formats:
- Classification: "Which is a rational / irrational / prime number?" or "Which statement is false?"
- Missing digit: "If 4★2,316 is divisible by 9 (or 11), find ★" — often the smallest or largest value.
- Pick the divisible number: four options, one divisible by 12, 24, 36, 72 or 88.
- Division algorithm: "Divisor 23, quotient 17, remainder 9; find the dividend." (23 × 17 + 9 = 400.)
- Remainders: of a large number, a product, a power, or by a factor of the original divisor.
- Largest / smallest multiple: "Largest 4-digit number divisible by 88." (9,999 ÷ 88 leaves 55 → 9,999 − 55 = 9,944.)
Difficulty: easy = one rule, one step (classify, test 3, 5 or 9); medium = missing digit for 9 or 11, composite divisors, finding a dividend; hard = remainders of products and powers, remainder by a factor, or two conditions together (divisible by both 8 and 9).
- Find the remainder when 7,345,963 is divided by 9.
Answer: Digit sum 37 → 37 ÷ 9 leaves 1. - A number leaves remainder 3 when divided by 5. What remainder does its square leave when divided by 5?
Answer: 3² = 9 → 9 ÷ 5 leaves 4. - Is 0 a natural number? Is it a whole number?
Answer: Not natural; yes, it is whole (and an integer). - Find the smallest 4-digit number divisible by 88.
Answer: 1,000 ÷ 88 leaves 32, so add 88 − 32 = 56 → 1,056.
- ✓- N ⊂ W ⊂ Z ⊂ Q ⊂ R; 0 is whole but not natural.
- ✓- 1 is neither prime nor composite; 2 is the only even prime; 25 primes up to 100.
- ✓- Digit sum for 3 and 9; last 1, 2, 3 digits for 2, 4, 8; alternating difference for 11.
- ✓- Composite divisor → co-prime factors only (24 = 3 × 8).
- ✓- Dividend = Divisor × Quotient + Remainder (remainder < divisor).
- ✓- For products and powers, work with remainders, not full numbers.
తెలుగు సారాంశం (Telugu summary)
సహజ సంఖ్యలు 1 నుంచి మొదలవుతాయి. వాటికి 0 కలిపితే పూర్ణాంకాలు, రుణ సంఖ్యలను కూడా కలిపితే పూర్ణ సంఖ్యలు వస్తాయి.
p/q రూపంలో (q ≠ 0) రాయగలిగే సంఖ్య అకరణీయ సంఖ్య; రాయలేనిది కరణీయ సంఖ్య (ఉదా: √2, π). ఈ రెండూ కలిసి వాస్తవ సంఖ్యలు.
1 ప్రధాన సంఖ్యా కాదు, సంయుక్త సంఖ్యా కాదు. 2 ఒక్కటే సరి ప్రధాన సంఖ్య. 0 సరి సంఖ్య.
3 మరియు 9 తో భాజనీయత కోసం అంకెల మొత్తం చూడండి. 2, 4, 8 కోసం చివరి 1, 2, 3 అంకెలు చూడండి.
11 తో భాజనీయత: బేసి స్థానాల అంకెల మొత్తం, సరి స్థానాల అంకెల మొత్తం మధ్య తేడా 0 లేదా 11 గుణిజం కావాలి.
24, 72 లాంటి సంయుక్త భాజకాలను పరస్పర ప్రధాన కారణాంకాలుగా (24 = 3 × 8) విడదీసి, రెండు పరీక్షలూ చేయండి.
విభాజ్యం = భాజకం × భాగఫలం + శేషం. శేషం ఎప్పుడూ భాజకం కంటే చిన్నదిగా ఉండాలి.
పెద్ద సంఖ్యల లబ్ధానికి శేషం కావాలంటే, ముందు ఒక్కో సంఖ్య శేషం తీసుకుని గుణించి, మళ్లీ శేషం తీసుకోండి.
ఘాతాల కోసం: (x + 1)ⁿ ని x తో భాగిస్తే శేషం ఎప్పుడూ 1.
Key terms (English — తెలుగు):
- Natural numbers (1, 2, 3, …) — సహజ సంఖ్యలు
- Whole numbers (0, 1, 2, …) — పూర్ణాంకాలు
- Integers (…, −1, 0, 1, …) — పూర్ణ సంఖ్యలు
- Rational numbers — అకరణీయ సంఖ్యలు
- Irrational numbers — కరణీయ సంఖ్యలు
- Real numbers — వాస్తవ సంఖ్యలు
- Prime number — ప్రధాన సంఖ్య
- Composite number — సంయుక్త సంఖ్య
- Even / Odd number — సరి / బేసి సంఖ్య
- Co-prime numbers — పరస్పర ప్రధాన సంఖ్యలు
- Divisibility — భాజనీయత
- Dividend — విభాజ్యం
- Divisor — భాజకం
- Quotient — భాగఫలం
- Remainder — శేషం
- Factor — కారణాంకం